Abstract
We provide a sharp generalization to the nonautonomous case of the well-known Kobayashi estimate for proximaliterates associated with maximal monotone operators. We then derive a bound for the distance between acontinuous-in-time trajectory, namely the solution to the differential inclusion $\dot{x} + A(t)x $∋ $ 0$, and thecorresponding proximal iterations. We also establish continuity properties with respect to time of the nonautonomousflow under simple assumptions by revealing their link with the function $t \mapsto A(t)$. Moreover, our sharperestimations allow us to derive equivalence results which are useful to compare the asymptotic behavior of thetrajectories defined by different evolution systems. We do so by extending a classical result of Passty to thenonautonomous setting.
Highlights
Motivated by either the existence or the algorithmic approximation of solutions to a differential inclusion problem of the type x + A(t)x ∋ 0, (1.1)where A(t) is a possibly time-dependent m-accretive operator with domain in a Banach space, several authors have considered some special implicit discretization schemes.In the autonomous case where A(t) ≡ A, Crandall and Liggett introduced in [8] the following limit: S(t)x0 = lim n→∞ I + t n A−n x0 = nl→im∞(JtA/n)nx0
The results by Pavel in [22] are closely related to the latter; the author presents some Kobayashi-type estimations and uses them to derive the existence of DS-limit solutions of the differential inclusion
Sometimes we identify A with its graph by writing [u, v] ∈ A for v ∈ Au
Summary
Motivated by either the existence or the algorithmic approximation of solutions to a differential inclusion problem of the type x + A(t)x ∋ 0,. Under some additional conditions on the operator-valued function t → A(t), Kobayasi et al gave in [17, Lemma 3.4] a first nonautonomous version of the original Kobayashi inequality They obtained important properties of the corresponding continuous dynamics by passing to the limit in an appropriate manner. The results by Pavel in [22] are closely related to the latter; the author presents some Kobayashi-type estimations and uses them to derive the existence of DS-limit solutions of the differential inclusion Their assumptions are very similar to ours but demand more precise information on the time-dependence. The converse is true in Hilbert space but not in general Banach spaces (see counterexample in [14])
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